Disjoint Covering of Bipartite Graphs with $s$-Clubs

For a positive integer , an -club in a graph is a set of vertices inducing a subgraph with diameter at most . As generalizations of cliques, -clubs offer a flexible model for real-world networks. This paper addresses the problems of partitioning and disjoint covering of vertices with -clubs on bipartite graphs. First we prove that for any fixed and fixed , determining whether the vertices of can be partitioned into at most disjoint -clubs is NP-complete even for bipartite graphs. Note that our NP-completeness result is stronger than the one in Abbas and Stewart (1999), as we assume that both and are constants and not part of the input. Additionally, we study the Maximum Disjoint -Club Covering problem (MAX-DCC( )), which aims to find a collection of vertex-disjoint -clubs (i.e. -clubs with at least vertices) that covers the maximum number of vertices in . We prove that it is NP-hard to achieve an approximation factor of for MAX-DCC( ) for any fixed and for MAX-DCC( ) for any fixed even for bipartite graphs. Previously, results were known only for MAX-DCC( ). Finally, we provide a polynomial-time algorithm for MAX-DCC( ) resolving an open problem from Dondi \textit{et al.} (2019).